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Chapter 12: Problem 42

Find the indicated derivative for the following functions. $$\partial z / \partial p, \text { where } z=x / y, x=p+q, \text { and } y=p-q$$

Short Answer

Expert verified
Answer: \(\frac{\partial z}{\partial p}=\frac{2q}{(p-q)^2}\)

Step by step solution

01

Express z in terms of p and q

Using equations 2 and 3, we can express z in terms of p and q as follows: \(z=\frac{x}{y}=\frac{p+q}{p-q}\) Now we have z as a function of p and q.
02

Apply the chain rule to find the partial derivative of z with respect to p

The partial derivative of z with respect to p can be found using the chain rule: \(\frac{\partial z}{\partial p}=\frac{\partial(\frac{p+q}{p-q})}{\partial p}\)
03

Differentiate the numerator with respect to p

Differentiating the numerator (p+q) with respect to p, we get: \(\frac{\partial(p + q)}{\partial p}=1\)
04

Differentiate the denominator with respect to p

Differentiating the denominator (p-q) with respect to p, we get: \(\frac{\partial(p - q)}{\partial p}=1\)
05

Apply the quotient rule for derivatives

Using the quotient rule for derivatives, we get: \(\frac{\partial z}{\partial p}=\frac{(1)(p-q) - (1)(p+q)}{(p-q)^2}=\frac{2q}{(p - q)^2}\) Therefore, the partial derivative of z with respect to p is: \(\frac{\partial z}{\partial p}=\frac{2q}{(p-q)^2}\)

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